multicolored subgraphs in an edge colored graph hung-lin fu department of applied mathematics nctu,...
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![Page 1: Multicolored Subgraphs in an Edge Colored Graph Hung-Lin Fu Department of Applied Mathematics NCTU, Hsin Chu, Taiwan 30050](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d6a5503460f94a48a44/html5/thumbnails/1.jpg)
Multicolored Subgraphs in Multicolored Subgraphs in an Edge Colored Graphan Edge Colored Graph
Multicolored Subgraphs in Multicolored Subgraphs in an Edge Colored Graphan Edge Colored Graph
Hung-Lin FuHung-Lin FuDepartment of Applied MathematicsDepartment of Applied Mathematics
NCTU, Hsin Chu, Taiwan 30050NCTU, Hsin Chu, Taiwan 30050
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Preliminaries
A (proper) k-edge coloring of a graph G is a mapping from E(G) into {1,…,k} (such that incident edges of G receive distinct colors).
A 3-edge coloring of 5-cycle
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Facts on Edge-Colorings
• Let G be a simple graph with maximum degree (G). Then, the minimum number of colors needed to properly color G, (G), is either (G) or (G) + 1. (Vizing’s Theorem)
• G is of class one if (G) = (G) and class two otherwise.
• Kn is of class one if and only if n is even.• Kn,n is of class one.
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Rainbow Subgraph• Let G be an edge-colored graph. Then a subgra
ph whose edges are of distinct colors is called a rainbow subgraph of G.
• It is also known as a heterochromatic subgraph or a multicolored subgraph.
• Note that we may consider the edge-coloring of the edge-colored graph which is not a proper edge-coloring.
• In this talk, all edge-colorings are proper edge-colorings. Therefore, a rainbow star can be found easily.
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Rainbow 1-factorTheorem (Woolbright and Fu, JCD 1998) In any (2m-1)-edge-colored K2m where m > 2,
there exists a rainbow 1-factor.
Conjecture (Fu) In any (2m-1)-edge-colored K2m, there exist 2
m-1 edge-disjoint rainbow 1-factors for integers m which are large enough.
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Theorem (Hatami and Shor, JCT(A) 2008) In any n-edge-colored Kn,n, there exists a ra
inbow matching of size larger than n – (11.053)(log n)2.
Conjecture (Ryser) In any n-edge-colored Kn,n, there exists a ra
inbow 1-factor if n is odd and there exists a rainbow matching of size n – 1 if n is even.
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•What if we can assign the edge-colorings?
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Room Squares• A Room square of side 2m-1 provides a (2m-1)-
edge-coloring of K2m such that 2m-1 edge- disjoint multicolored 1-factors exist.
35 17 28 46 26 48 15 37 13 57 68 24 47 16 38 25 58 23 14 67 12 78 56 34 36 45 27 18
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Orthogonal Latin Squares
• A Latin square of order n corresponds to an n-edge-coloring of Kn,n.
• A Latin square of order n with an orthogonal mate provides n edge-disjoint multicolored 1-factors of Kn,n.
1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1
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Multicolored Subgraph• Conjecture Given an (n-1)-edge-colore
d Kn for even n > n0, a multicolored Hamiltonian path exists. (By whom?)
• Problem (Fu and Woolbright) Find “a” longest multicolored path in
a (Kn)-edge-colored Kn.
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Brualdi-Hollingsworth’s Conjecture
If m>2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m multicolored spanning trees.
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Multicolored Tree Parallelism
• K2m admits a multicolored tree parallelism (MTP) if there exists a proper (2m-1)-edge-coloring of K2m for which all edges can be partitioned into m isomorphic multicolored spanning trees.
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K6 admits an MTP T1 T2 T3
Color 1: 35 46 12 Color 2: 24 15 36 Color 3: 25 34 16 Color 4: 26 13 45 Color 5: 14 23 56
2
5 46
13
T1
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Two Conjectures on MTP
Constantine’s Weak Conjecture For any natural number m > 2,
there exists a (2m-1)-edge-coloring of K2m for which K2m can be decomposed into m multicolored isomorphic spanning trees.
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Constantine’s Strong Conjecture
If m > 2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m isomorphic multicolored spanning trees. (Three, so far!)
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Theorem (Akbari, Alipour, Fu and Lo, SIAM DM)
For m is an odd positive integer, then K2m admits an MTP.
Fact. Constantine’s Weak Conjecture is true.
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Edge-colored Kn, n is Odd
• It is well known that Kn is of class 2 when n is odd, i. e. the chromatic index of Kn is n.
• In order to find multicolored parallelism, each subgraph has to be of size n. The best candidate is therefore the Hamiltonian cycles of Kn. (Unicyclic spanning subgraphs are also great!)
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• K2m+1 admits a multicolored Hamiltonian
cycle parallelism (MHCP) if there exists a
proper (2m+1)-edge-coloring of K2m+1 for
which all edges can be partitioned into m multicolored Hamiltonian cycles.
MHCP
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• Theorem (Constantine, SIAM DM)If n is an odd prime, then Kn admits an MHCP.
• Conjecture Kn admits an MHCP for each odd integer n.
The Existence
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Lemma Let v be a composite odd integer and n is the smallest prime which is a factor of v, say v = mn. If Km admits an MHCP, then Km(n) admits an MHCP.
Theorem Kv admits an MHCP for each odd integer v.
MHCP (Fu and Lo, DM 2009)
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• Let μ be an arbitrary (2m-1)-edge-coloring of K2m. Then
there exist three isomorphic multicolored spanning trees
in K2m for m > 2.
• Joint work with Y.H. Lo.
Can we do it if the edge-coloring is given?
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Observation
• If the edge-coloring is arbitrarily given, then finding MTP is going to be very difficult.
• If we drop “isomorphism” requirement for the above case, then may be we can find many multicolored spanning trees of K2m?
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Problem: How many multicolored spanning
trees of an edge-colored K2m can we find if m is
getting larger?
•Guess? Of course, the best result is m.
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Joint work with Y.H. Lo• Theorem For any proper (2m-1)-
edge-coloring of K2m, there exist around m1/2 mutually edge-disjoint multicolored spanning trees.
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Definitionφ is a (2m-1)-edge-coloring of K2m, and φ(xy) = c.
Define 1.φx-1 (c) = y and φy
-1 (c) = x
2. xy = x‹c› = y‹c› x
y
1
2
3
4
5
= φx-1 (4)
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Assume T is a multicolored spanning tree of K2m with
two leaves x1, x2. Let the edges incident to x1 and x2 be
e1 and e2 respectively..
Define T[x1,x2] = T – {e1,e2} + {x1‹c2›, x2‹c1›}.
Definition
3
4 u
v
1
23
4
5
T
u
v
1
2
5
3
4
T[u,v]
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Sketch proof1. Pick any two vertices x∞, x1, let T∞ and T1 be the stars
with centers x∞, x1, respectively.
x∞ x1
T∞
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2. Pick x2, u, v1, and let the colors be as follows.
x∞ x1
v1x2u
1 2c1
c2
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2. Pick x2, u, v1, and let the colors be as follows.
.
x∞ x1
v1x2u
1 2c1
c2
2 1
u1
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2. Pick x2, u, v1, and let the colors be as follows.
Find and . Redefine T1 = T1[x2,v1]
x∞ x1
v1x2u2 1
2
11(2)x u
1
1(1)v
u1
T1
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2. Pick x2, u, v1, and let the colors be as follows.
Find and . Define
x∞ x1
x2u2
1
22 1T [ , ]xS u u2
11(2)x u
1
1(1)v
u1
T2
c1
c2
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2. The structure of T∞ so far.
x∞ x1
T∞
x2
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2. The structure of T1 so far.
x∞ x1
T1
x2
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2. The structure of T2 so far.
x∞ x1
T2
x2
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3. Choose x3, u, v1, v2 and let the colors be as follows.
x∞ x1 x2
v1x3u v2
1 2c1c
2
3 4
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3. Choose x3, u, v1, v2 and let the colors be as follows.
Redefine T1 = T1[x3, v1].
x∞ x1
T1
x2
v1x313u v2u1
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3. Choose x3, u, v1, v2 and let the colors be as follows.
Redefine T2 = T2[x3, v2].
x∞ x1
T2
x2
v1x3u v2
24
u2
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3. Choose x3, u, v1, v2 and let the colors be as follows.
Define
x∞ x1
T3
x2
x334
33 1 2T [ , ]xS u u
u1 u2
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• Adjust T1, T2, T3 to add an extra T4.
• ………
• ………until the n-th tree is found.
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Property: 1. Ti and Tj are edge-disjoint for i ≠ j ≠ ∞.
2. In each Ti, i ≠ ∞, x∞ is a leaf which is of distant 1 from
its center xi.
3. The number of trees n is determined by way of m.
T1, T2, T3, …, Tn are the desired trees.
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Corollary For any proper (2m-1)-edge-coloring of K
2m-1, there exist around m1/2 mutually edge-dis
join multicolored unicyclic spanning subgraphs.
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1. Adding an extra vertex, named x∞, to form a (2m-1)-edge-colored K2m.
2. Apply Theorem 1 to construct T∞, T1, T2, …
3. Drop x∞.
4. Adding one specific edge colored the missing color φ(x∞xi) to each Ti, for i =1,2,...,
Sketch proof
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Don’t Stop!